Difference between Ln and Log: Definitions and Properties

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Difference between ln and log is that ln is defined for base e while log is defined for base 10 i.e., log_e = ln. Ln is also called natural log. In mathematics, logarithm is defined as the inverse function of exponentiation. In simpler words, logarithm is the power (exponent) to which a base has to be raised to attain a given number i.e., the number of times a base should be multiplied by itself to attain a given number. Logarithms help to cover a large range of data/values in a crisp manner. Due to this reason, it has numerous applications in the field of mathematics, physics and chemistry.

Read More: Value of e

Table of Content

Definition of ln
Definition of log
Difference between ln and log
Application of Ln and Log
Things to Remember
Sample Questions

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Definition of Ln

Ln is known as the natural logarithm i.e., logarithm with the base e (an irrational and a transcendental number) which is approximately equal to 2.7183. It is referred to by ln x or log_e x. Here, e is the Euler’s constant.

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log_e x = ln x = e^x = y

A table representing the ln values of 1 to 10 is:

ln (1)	0
ln (2)	0.6931
ln (3)	1.0986
ln (4)	1.3863
ln (5)	1.6094
ln (6)	1.7918
ln (7)	1.9459
ln (8)	2.0794
ln (9)	2.1972
ln (10)	2.3026

Rules of Ln

Power Rule:

Ln (xy) = y * ln (x)

Example: ln (53) = 3 * ln (5)

Quotient Rule:

Ln (x/y) = ln (x) – ln (y)

Example: ln (7/3) = ln (7) – ln (3)

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Reciprocal Rule:

Ln (1/x) = - ln (x)

Example: ln (1/5) = - ln (5)

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Value of log 0	Real Valued Function
Geometric Progression	Geometric Mean

Definition of Log

Log is known as the logarithm which is defined for the base 10. It is also called common logarithm.

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Log₁₀ y = x

10x = y

A table representing the log values of 1 to 10 is:

log (1)	0
log (2)	0.3010
log (3)	0.4771
log (4)	0.6020
log (5)	0.6989
log (6)	0.7781
log (7)	0.8450
log (8)	0.9030
log (9)	0.9542
log (10)	1

Properties of Log:

Log (mn) = log m + log n

Example: log (3*5) = log (3) + log (5)

Log (m/n) = log m – log n

Example: log (3/2) = log (3) – log (2)

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Log (mn) = n log m

Example: log (52) = 2 * log (5)

Log_b m = log_a m/log_a b

Example: log₃ 8 = log₂ 8 * log_a 2

Difference between Ln and Log

S. No.	Ln	Log
1.	Ln is defined as the logarithm with base e	Log is defined as the logarithm with base 10
2.	It is also known as natural logarithm	It is also known as common logarithm
3.	It is represented as log_e x	It is represented as log₁₀ x
4.	Its exponential form is e^x= y	Its exponential form is 10^x = y
5.	General statement for exponential logarithm is “By what number must e be raised to attain y”	General statement for exponential logarithm is “By what number must 10 be raised to attain y”
6.	It is mathematically represented as log base e	It is mathematically represented as log base 10
7.	It finds less application in physics	It finds more application in physics

Application of Ln and Log

The real life applications are:

Logarithms are used in the Richter Scale for earthquakes.
It is also used in the detection of the password strength
For data scientists, logarithms help in seeing the patterns in bulk data
Log odds play a pivotal role in logistic regression

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Sequence and Series	Arithmetic Progression

Points to Remember

The general form of a logarithm is:

log_a (y) = x

The above expression can also be denoted as:

a^x = y

log_e x = ln x = e^y = y
log_a 1 = 0
log_a a = 1
Ln (x) = 2.303 * log (x)
Log (x) = ln (x) / 2.303

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Sample Questions

Ques 1. Define logarithms. [1 mark]

Ans. Logarithm of a given number with base x is the power to which x should be raised to attain the given number.

a^x = y

log_a y = x

Ques 2. Define natural and common logarithms. [2 mark]

Ans. Natural logarithm is a logarithm with the base e(Euler’s constant which is approximately equal to 2.7183) i.e., log_e x = ln x = e^x

Common logarithm is a logarithm with the base 10 i.e., log₁₀ x

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Ques 3. What is the value of log 9 when log 27 is equal to 1.431? [3 mark]

Ans. Given,

Log 27 = 1.431

Log (33) = 1.431

3 * log 3 = 1.431

Log 3 = 1.431/3

Log 3 = 0.447

Log 9 = log (32)

Log 9 = 2 * log 3

Log 9 = 2 * 0.447

Log 9 = 0.954

Ques 4. Given that the value of log₁₀ 2 = 0.3010, what is the value of log₁₀ 80? [3 mark]

Ans. Log₁₀ 80 = log₁₀ (8 * 10)

Log₁₀ 8 + log₁₀ 10

Log₁₀ (23) + 1

3 * log 2 + 1

3 * 0.3010 + 1

0.9030 + 1

Hence, the value of log10 80 is 1.9030

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Ques 5. If the value of log₁₀ 5 + log₁₀ (5x + 1) = log₁₀ (x + 5) + 1, what is the value of x? [4 mark]

Ans. Given,

log₁₀ 5 + log₁₀ (5x + 1) = log₁₀ (x + 5) + 1

log₁₀ 5 + log₁₀ (5x + 1) = log₁₀ (x + 5) + log₁₀ 10

log₁₀ [5 (5x + 1)] = log₁₀ [10(x + 5)]

5(5x + 1) = 10(x + 5)

5x + 1 = 2x + 10

3x = 9

Hence, the value of x is 3.

Ques 6. If the value of log₁₀ 7 = a, then log₁₀ (1/70) is equal to? [3 mark]

Ans. log₁₀ (1/70) = log₁₀ 1 – log₁₀ 70

= - log₁₀ (7 * 10)

= - (log₁₀ 7 + log₁₀ 10)

= - (a + 1)

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Ques 7. How can you convert natural log to common log? [2 mark]

Ans. To convert a natural logarithmic value to common logarithmic value, the following formula can be used:

Ln (x) = log (x) ÷ log (2.71828)

Ques 8. Find the value of (1/log₃ 60 + 1/log₄ 60 + 1/log₅ 60) [3 mark]

Ans. Given,

(1/log₃ 60 + 1/log₄ 60 + 1/log₅ 60)

= log₆₀ (3 * 4 * 5)

= log₆₀ (60)

Hence, the answer to this question is 1.

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